MACLAURIN AND TAYLOR SERIES Elementary Functions Most of the functions that we have been dealing in this class are so called elementary functions. Definition. An elementary function is a function that are polynomials, power functions, ratio- nal functions, trigonometric funcitions, inverse trigonometric functions, exponential functions, logarithmic functions, and all functions that can be obtained from these functions by addition, substraction, multiplication, division and composition. For example, the function √ ex log15(sin(e ) f(x) = ln(ln(ln(ln(cos ex + cosh x)))) − xπ is an elementary function. Remark. If f is an elementary function, then f (x) is also an elementary function. However, f(x) dx need not be an elemnentary function. For instance, 2 ex dx is not an elementary function. Question 1. Read page 345, Can We Integrate all Continuous Functions? Taylor Series In the previous section, we found a power series representation for certain family of functions. For example, ∞ 1 n − − = x ; 1 <x<1 1 x n=0 The natural question to ask is whether all power series represent familiar elementary functions. The answer is no, there are many power series that do not represent familiar elementary functions (polynomial functions, rational functions, algebraic functions, trigonometric functions, inverse trigonometric functions, exponential functions, logarithmic functions, etc.). Example. The power series ∞ xn 2 n=0 n +1 cannot be expressed as an elementary function. But there is a large class of power series that do represent elementary functions. There are the series that we study in this chapter. Such series are called Maclaurin series or Taylor series. 1 2 MACLAURIN AND TAYLOR SERIES Theorem 0.1. If f has a power series representation at a, that is, if ∞ n f(x)= cn(x − a) |x − a| <R n=0 then it’s coefficients are given by the formula f (n)(a) cn = n! In other words, if f has a power series expansion at a, then it must be of the form ∞ f (n)(a) f(x)= (x − a)n n=0 n! This series is called the Taylor series of function about a (or centered at a). Definition (Maclaurin Series). In the special case that a =0,theseries ∞ n f(x)= cnx n=0 f (n)(0) where cn = ,iscalledtheMaclaurinseriesof the function f(x). n! Remark. If f can be represented as a power series about a,thenf is equal to the sum of it’s Taylor series. However, there exist functions that are not equal to the sum of their Taylor Series. Important Maclaurin Series The following Maclaurin series are very important and you are expected to remember them: ∞ 1 2 3 ··· n − (1) − =1+x + x + x + = x , 1 <x<1. 1 x n=0 ∞ 1 (2) =1− x + x2 − x3 + ···= (−1)nxn, −1 <x<1. 1+x n=0 ∞ x2 x3 xn (3) ex =1+x + + + ···= , x in (−∞, ∞). 2! 3! n=0 n! ∞ x2 x4 x6 (−1)n · x2n (4) cos(x)=1− + − + ···= , x in (−∞, ∞). 2! 4! 6! n=0 (2n)! ∞ x3 x5 x7 (−1)n · x2n+1 (5) sin(x)=x − + − + ···= , x in (−∞, ∞) 3! 5! 7! n=0 (2n +1)! Remark. To avoid confusion between the series for sin x and cos x, remember that: • sin x is an odd function, that is, sin(−x)=− sin x,sotheseriesforsinx has only odd exponents. • cos x is an even function, that is cos(−x)=cosx,sotheseriesforcosx has only even exponents. Question 2. Verify the Maclaurin series representation of ex. MACLAURIN AND TAYLOR SERIES 3 Question 3. Verify the Maclaurin series representation of sin x. Question 4. Use you answer in the previous question to verify the Maclaurin series represen- tation of cos x. 4 MACLAURIN AND TAYLOR SERIES Question 5. Find the Taylor series for ln x about x = 1, and find the interval of convergence. Question 6. Express x3 in powers of x − 3. MACLAURIN AND TAYLOR SERIES 5 Question 7. Find the Maclaurin series for cos 2x. Algebraic Operations. Other than the differentiation and integration of power series, there are other algebraic operations of power series that you will need to know: ∞ ∞ n n Let f(x)= anx and g(x)= bnx . Then, within a common interval of convergence: n=0 n=0 ∞ n (1) f(x)+g(x)= (an + bn)x . n=0 ∞ n (2) f(x) − g(x)= (an − bn)x n=0 ∞ n n (3) f(x) · g(x)= cnx ,wherecn = a0bn + a1bn−1 + ···an−1b1 + anb0 = ak bn−k.This n=0 k=0 ∞ ∞ n n is called the Cauchy product of anx and bnx . This is really a term by term n=0 n=0 multiplication of polynomials, except, there are infinitely many terms. ∞ ∞ n n 2 2 anx · bnx =(a0 + a1x + a2x + ···) · (b0 + b1x + b2x + ···) n=0 n=0 2 3 ··· = a0b 0 +(a1b0 + a0b1 ) x +(a0b2 + a1b1 + a2b0 ) x +(a0b3 + a1b2 + a2b1 + a3b0 ) x + c0 c1 c2 c3 6 MACLAURIN AND TAYLOR SERIES Question 8. Find the Maclaurin series for cosh x, and its interval of convergence. 1 Hint: cosh x = (ex + e−x). 2 Question 9. Find the first 3 non-zero terms of the Maclaurin series of f(x)=ex cos x,by multiplying the two series. Verify your answers using the direct method (definition of a Maclaurin series). MACLAURIN AND TAYLOR SERIES 7 Series Manipulation Before we proceed with the next section, we will need to discuss an important aspect of dealing with series, which is manipulating the index of the series. You can reindex a series by changing the starting place of the series. This might change the form of an, but will not change the value of the series. The trick to remember when reindexing is that: When you decrease the starting place, you have to increase every n in an by the same amount. For example, ∞ ∞ xn+1 x(n+1)+1 = n=1 (n +1)! n=0 ((n + 1) + 1)! You need to be careful when changing the index, remember there is a ”hidden parentheses around each n. For instance, ∞ ∞ ∞ nx2n+1 (n +3)x2(n+3)+1 (n +3)x2n+7 = = n=3 (2n)! n=0 (2(n + 3))! n=0 (2n +6)! A few other things you might wanna keep in mind when manipulating a series is that you can factor expressions not involving n in or out of the series. Finally, you might have to add or substract missing terms: For example, ∞ ∞ 4 ∞ xn+1 xn+1 xn+1 xn+1 x1 x2 x3 x4 x5 = − = − + + + + n=5 n! n=0 n! n=0 n! n=0 n! 0! 1! 2! 3! 4! Question 10. Simplify the power series by manipulating the index of the series ∞ n−2 ∞ n+1 − n x − − n−1 x ( 1) − ( 1) n=3 (n 3)! n=0 n! 8 MACLAURIN AND TAYLOR SERIES Application of Series Many differential equations can’t be solved explicity in terms of elementary functions. But it is important to be able to solve such equations. In such cases, we use method of power series; that is, we look for solutions of the form ∞ n 2 3 y = f(x)= cnx = c0 + c1x + c2x + c3x + ··· n=0 Question 11. Use power series to solve the equation y − y =0. MACLAURIN AND TAYLOR SERIES 9 Estimating the sum of a series We have now spent a lot of time determining the convergence of a series, however, with the exception of geometric and telescoping series, we have not talked about finding the value of a series. This is usually a very difficult thing to do but what we can do instead is estimate the value of a series. It makes no sense to talk about the value of a series that doesnt converge and so we will be assuming that the series were working with are convergent. ∞ Assume that the series an converges to some number s. This means that the sequence of n=1 partial sums {sn} converges to s, that is, lim sn = s. n→∞ The sequence of partial sum converges, means that we can make the partial sums, sn,asclose to s as we want simply by taking n large enough. (Hope you remember the idea behind what it means for a sequence to be convergent). We will take a partial sum and use that as an estimation of the value of the series, that is, s is approximately equal to sn (s ≈ sn). The estimation will be of no use to us unless we can estimate the accuracy of approximation. To do that consider Rn = |s − sn|, which we will call the remainder. The remainder tells us the difference, or error, between the exact value of the series and the value of the partial sum that we are using as the estimation of the value of the series. Question 12. Express Rn in-terms of an. (Hint: Rn is the tail of the series) For an alternating series that is convergent, the size of the error is smaller than bn+1,which is the absolute value of the first neglected term. n−1 Theorem 0.2 (Alternating Series Estimation Theorem). If s = (−1) bn is the sum of an alternating series that satisfies a) 0 ≤ bn+1 ≤ bn. b) lim bn =0. n→∞ Then, |Rn| = |s − sn|≤bn+1. 10 MACLAURIN AND TAYLOR SERIES ∞ (−1)n+1 Question 13. Find the sum of the series 5 correct to 4 decimal places. n=1 n The study of series comes from Newton’s idea of representing functions as sums of infinite series. Now that we have a good foundation is series, we will use this idea to study functions which are not elementary.